Symbolic dynamics and the category of graphs

Terrence Bisson and Aristide Tsemo

Symbolic dynamics is partly the study of walks in a directed graph. By a
walk, here we mean a morphism to the graph from the Cayley graph of the
monoid of non-negative integers. Sets of these walks are also important in
other areas, such as stochastic processes, automata, combinatorial group
theory, $C^*$-algebras, etc. We put a Quillen model structure on the
category of directed graphs, for which the weak equivalences are those
graph morphisms which induce bijections on the set of walks. We determine
the resulting homotopy category. We also introduce a "finite-level"
homotopy category which respects the natural topology on the set of walks.
To each graph we associate a basal graph, well defined up to isomorphism.
We show that the basal graph is a homotopy invariant for our model
structure, and that it is a finer invariant than the zeta series of a
finite graph. We also show that, for finite walkable graphs, if $B$ is
basal and separated then the walk spaces for $X$ and $B$ are topologically
conjugate if and only if $X$ and $B$ are homotopically equivalent for our
model structure.